Hydrodynamic Lubrication 2009 Part 13 pot

210 9 Turbulent Lubrication Fig. 9.4a,b. Results of calculation of the turbulent coefficients [33] for equal pressure gradients in the x and z directions (a) and for different pressure gradients (b) 1/G x = k x = 12(1 + α x R h n x ), 1/G z = k z = 12(1 + α z R h n z ) (9.53) where α x = 0.00116,α z = 0.00120, n x = 0.916, n z = 0.854 These expressions were obtained using Eq. 9.33 as a coefficient of modification for the mixing length. R h is the local Reynolds number. If R h is sufficiently small, G x = G z = 1/12 is obtained from the above, and therefore Eqs. 9.51 and 9.52 are in agreement with Reynolds’ equation in the case of laminar flow. 9.4 Comparison of Analyses Using the Mixing Length Model with Experiments 211 9.4 Comparison of Analyses Using the Mixing Length Model with Experiments In this section, some results of analyses of turbulent fluid film seals based on the mixing length model will be compared with experimental results. 9.4.1 Turbulent Static Characteristics of Fluid Film Seals Fig. 9.5. Fluid film seal [35] The turbulent static characteristics of a fluid film seal as shown in Fig. 9.5 are consid- ered (Kaneko et al. [32] [35]). The turbulent Reynolds’ equation (Eq. 9.51 or 9.52) is applied to the film of a seal and is solved by using the finite difference method. In the case of a fluid seal, a big pressure difference exists between the two ends, and a pressure loss takes place at the high pressure end where the liquid flows into the seal clearance. This is taken into consideration in the boundary conditions. The pressure loss is assumed to be expressed as follows when an average inflow velocity in the axial direction is w m : ∆p = C L ρw m 2 2 (9.54) where C L is a pressure loss coefficient, which, according to experiments, is given as follows: C L = −R 0 /2900 + 2.57, R 0 = w m h/ν (9.55) with a proviso that C L = constant when R 0 < 1000. (9.56) 212 9 Turbulent Lubrication R 0 is the local Reynolds’ number in the axial direction. When the journal and seal are eccentric, the average flow velocity w m changes with the position on the seal circumference, and hence the pressure loss also changes with the position on the circumference. To solve Eq. 9.51 for the pressure distribu- tion, the pressure losses on the seal circumference must be known as a boundary condition, and the pressure distribution must be known beforehand to know the pres- sure loss because it is a function of inflow velocity. Thus, an iterative calculation is needed. If the pressure distribution can be found in this way, the load capacity, the journal center loci, and so on of a fluid film seal in a turbulent condition can be obtained. The method of calculation is the same as that for a journal bearing. In the case of a fluid film seal, the pressure difference p d acting on the seal affects these characteristics. Fig. 9.6. Load capacity and eccentricity ratio in a fluid film seal. Chained line,laminarflow; solid lines, turbulent flow (theoretical values). Sysmbols show experimental values: diamonds, pressure difference p d = 0.1MPa;circles, p d = 0.3MPa;triangles, p d = 0.7 MPa [35] Figure 9.6 shows the relationship between the seal pressure (load capacity per unit area) p m and the eccentricity ratio κ at a rotating speed of N = 4080, with a pres- sure difference p d between the two ends of the seal as a parameter. The experimental values are close to the theoretical values in the case of turbulent flow. 9.4 Comparison of Analyses Using the Mixing Length Model with Experiments 213 It is seen in Fig. 9.6 that the seal pressure p m increases with an increase in the eccentricity ratio. This is quite natural. On the other hand, it is very interesting to note that the seal pressure p m increases with the increase in the pressure difference p d . When a journal is in an eccentric position in a seal, the inflow velocity of the fluid and hence the pressure loss is large at the circumferential position where the seal clearance is large, whereas the pressure loss is small, in contrast, at the position where the seal clearance is small. The difference of the pressure loss yields a static bearing effect in the oil film, and this is added to the oil film force produced by journal rotation. The larger the pressure difference at the two ends of a seal p d ,the larger the static bearing effect and the load capacity (per unit area) p m will be. Fig. 9.7. Loci of journal center in a fluid film seal [35] Figure 9.7 shows the calculated loci of the journal center with pressure difference p d and journal rotating speed N as parameters. In this case, the loci of the journal center are closer to the vertical line (the direction of loading) for larger p d ,orfor smaller N. This is because the static bearing effect due to the pressure difference p d contributes mainly to the P κ component of the oil film force but does not contribute to the P θ component. Consequently, the larger the pressure difference is, the smaller the attitude angle is. When the rotating speed is low, the static bearing effect becomes relatively large, hence the attitude angle becomes small. 9.4.2 Turbulent Dynamic Characteristics of Fluid Film Seals The elastic and damping coefficients of a turbulent fluid film seal are considered. In this case also, the pressure difference between the two ends of the seal affects these coefficients (Kaneko et al. [32] [36] [37]). 214 9 Turbulent Lubrication In reference to the coordinate axes x and y of Fig. 9.8, elastic coefficient K ij and damping coefficient C ij are defined as Eq. 9.57 using the oil film forces P x and P y , where the positive direction of P x and P y are taken in the direction of −x and −y. The subscript 0 denotes the static equilibrium position. To obtain P x and P y , first solve the turbulent Reynolds’ equation (Eq. 9.51) for the pressure, multiply the obtained pressure by cos φ and sin φ, integrate these with respect to φ to obtain P κ and P θ (see Fig. 9.5), and finally transform these into P x and P y . K xx = ∂P x ∂x      0 , K xy = ∂P x ∂y      0 , K yx = ∂P y ∂x       0 , K yy = ∂P y ∂y       0 C xx = ∂P x ∂ ˙x      0 , C xy = ∂P x ∂˙y      0 , C yx = ∂P y ∂ ˙x       0 , C yy = ∂P y ∂˙y       0 (9.57) Fig. 9.8. Axes of coordinates (horizontal, vertical) Figure 9.9 shows three examples of the above-mentioned elastic coefficients and a damping coefficient K xx , K xy , and C xx as functions of eccentricity ratio κ 0 with P d as a parameter. The figure shows that the larger the pressure difference is, the larger the value of these constants (absolute values) becomes. Further, the value of K xx for laminar flow is larger than that in the case of turbulent flow. The same can be said of K yy , although the data are not shown. The calculation conditions are as follows: D = 70 mm, L = 35 mm, c = 0.175 mm, N = 4000 rpm, µ = 1.44 mPa·s, the axial Reynolds’ number R a = w m c/ν = 767 – 2540 (w m is the axial average velocity), the circumferential Reynolds’ number R ω = Rωc/ν = 1418. 9.5 Turbulent Lubrication Theory Using the k-ε Model In the case of a journal bearing in which the eccentricity ratio of the journal is large, the pressure gradient in the oil film is large. The mixing length used in the mixing 9.5 Turbulent Lubrication Theory Using the k-ε Model 215 Fig. 9.9. Spring and damping coefficients of a fluid film seal [36] length model is usually determined experimentally for small pressure gradients, but it changes with pressure gradient. Therefore, it is more reasonable to use the k-ε model, which is less affected by pressure gradient, for analyses of a turbulent bearing with a large eccentricity ratio. An analysis based on the k-ε model will be described below (Kato and Hori [31]). 9.5.1 Application of the k-ε Model to an Oil Film In the oil film of a bearing, especially in the neighborhood of the wall surface, the tur- bulent Reynolds’ number R t is relatively low. The low-Reynolds’ number k-ε model, which is suitable in such a case, was introduced by Jones and Launder [21] [22] as stated in the previous section. The transport equations (Eqs. 9.22 and 9.23) for k and ε given by them were improved later by Hassid and Poreh [25]. We use Hassid and Poreh’s model here. This model can be applied to cases in which the Toms effect (described later) appears [34]. First, the turbulent energy k and the turbulent loss ε are defined as follows: k = 1 2 u i  u i  = 1 2  u  2 +   2 + w  2  (9.58) ε = ν ∂u i  ∂x j ∂u i  ∂x j = ν  ∂u  ∂x  2 + ···+  ∂w  ∂z  2 (9.59) The following two-dimensional equations, which are extentions of the one- dimensional equations by Hassid and Poreh, will be used as transport equations for k and ε: 216 9 Turbulent Lubrication Dk Dt = ∂ ∂y  ν + ν t σ k  ∂k ∂y  − u    ∂u ∂y −   w  ∂w ∂y − ε − 2νk b 2 (9.60) Dε Dt = ∂ ∂y  ν + ν t σ ε  ∂ε ∂y  −C ε1  u    ∂u ∂y +   w  ∂w ∂y  ε k −C ε2  1 −0.3exp(−R t 2 )  ε 2 k − 2ν  ∂ε 1/2 ∂y  2 (9.61) where b = min(y, h −y) σ k = 1,σ ε = 1.3, C ε1 = 1.45, C ε2 = 2.0 R t = k 2 /(εν) In Eqs. 9.60 and 9.61, ε denotes the isotropic part of the turbulent loss and 2νk/b 2 the anisotropic part. The idea of separating the turbulent loss in this way was pro- posed by Jones and Launder, and the following simple boundary conditions for ε has thereby become possible: ε = 0 at the wall surface (9.62) Further, the boundary conditions for k is assumed to be: k = 0 at the wall surface (9.63) According to Hassid and Poreh, the turbulent dynamic viscosity coefficient ν t is given as follows by using the above-mentioned k and ε: ν t = C m k 2 ε  1 −exp(−A d R t )  (9.64) However, the following equation, which contains a correction factor C d , will be used here, based on Laufer’s experiments on Couette flows [4]: ν t = C m k 2 ε  1 −C d exp(−A d R t )  (9.65) where C m = 0.09, A d = 1.5 ×10 −3 , C d = 0.95 Although a strong Couette flow in the circumferential direction and a weak pres- sure flow in the axial direction are expected to exist in a bearing, it is assumed here that Eq. 9.65 can be used in both the circumferential and the axial directions, based on the fact that the correction factor C d = 0.95 is close to 1. 9.5.2 Turbulent Reynolds’ Equation The time-average equation of motion of an incompressible fluid containing Reynolds’ stress in a two-dimensional case can be given by Eqs. 9.8 and 9.9. In general, it can be written in a tensor expression as follows: 9.5 Turbulent Lubrication Theory Using the k-ε Model 217 ρ  ∂u i ∂t + u j ∂u i ∂x j  = − ∂p ∂x i + ∂ ∂x j  µ ∂u i ∂x j − ρ u i  u  j  (9.66) (i, j = 1, 2, 3; summation is taken over all values of j) The equations in rectangular coordinates (x, y, z) can be obtained by the substitution of variables such as x = x 1 , y = x 2 , z = x 3 , u = u 1 ,  = u 2 , and w = u 3 , where x, y, and z are the coordinates in the circumferential direction, across the film thickness, and in the axial directions; u and u  (and similar) express the static (time-average) parts of the flow velocity and the fluctuations about it, respectively. Considering a sufficiently thin lubricating film, let us make the following as- sumptions. 1. The left-hand side (inertia term) of Eq. 9.66 is negligible. 2. In Eq. 9.66, the derivatives of Reynolds’ stresses with respect to x and z can be neglected compared with that with respect to y. 3. The normal components of Reynolds’ stress (components i= j) can be neglected. 4. −ρ u    and −ρ  w  can be expressed as follows with a turbulent viscosity coeffi- cient ν t , which is common to the x and z directions: −ρ u    = ρν t ∂u ∂y (9.67) −ρ   w  = ρν t ∂w ∂y (9.68) Under these assumptions, a turbulent lubrication equation is derived from Eq. 9.66. First, disregarding the left-hand side of Eq. 9.66 from assumption 1, then sub- stituting Eqs. 9.67 and 9.68 of assumption 4 into this leads to the following equations: ∂p ∂x = ρ ∂ ∂y  (ν + ν t ) ∂u ∂y  (9.69) ∂p ∂z = ρ ∂ ∂y  (ν + ν t ) ∂w ∂y  (9.70) where ∂p/∂y = 0 is omitted. Integrate Eqs. 9.69 and 9.70 twice with respect to y under the boundary conditions u = U 1 , w = 0aty = 0 (9.71) u = w = 0aty = h (9.72) to obtain u and w, respectively. Substituting these into the continuity equation  h 0 ∂u ∂x dy +  h 0 ∂w ∂z dy = 0 (9.73) gives the following turbulent Reynolds’ equation, with G x = G z = G: ∂ ∂x  G ∂p ∂x  + ∂ ∂z  G ∂p ∂z  = U 1 ∂F ∂x (9.74) 218 9 Turbulent Lubrication where G =  h 0  y 0 dy dy ν + ν t  h 0 ydy ν + ν t  h 0 dy ν + ν t −  h 0  y 0 ydydy ρ(ν + ν t ) (9.75) F = h −  h 0  y 0 dy dy ν + ν t  h 0 dy ν + ν t (9.76) If Eq. 9.65 is used, the five equations, Eqs. 9.60, 9.61, 9.69, 9.70, and 9.74 form a closed set of equations with respect to the five unknowns k, ε, u, w, and p.Un- knowns such as the fluctuations in the velocity are not included in the equations. Therefore, by solving these five equations simultaneously, the above five unknowns will be obtained. It is assumed here that the left-hand side of Eqs. 9.60 and 9.61 (time variations of k and ε along the streamline) can be disregarded, considering the stationary state: Dk Dt = 0, Dε Dt = 0 Turbulent lubrication problems can thus be solved. Approximate solutions are possible when the left-hand side (inertia term) of Eq. 9.66 cannot be disregarded [28]. 9.6 Comparison of Analyses Using the k-ε Model with Experiments In this section, some examples of comparisons of theoretical analyses of a turbulent bearing by the k-ε model and experiments will be shown (Kato and Hori [31]). In the- oretical calculations, Eqs. 9.60, 9.61, and 9.74 are solved simultaneously under the boundary conditions given in Eqs. 9.71, 9.72, 9.62, 9.63 and the following boundary condition concerning pressure: p = 0atθ = 0,πand at the bearing ends. (9.77) The procedure for numerical calculations is as follows. Assume suitable initial profiles of k and ε, calculate G and F of Eqs. 9.75 and 9.76 and then obtain the pressure distribution by applying the finite element method to the turbulent lubrica- tion equation (Eq. 9.74). Next, calculate the flow velocity distributions u and w from the pressure distribution by using Eqs. 9.69 and 9.70 and the boundary conditions Eqs. 9.71 and 9.72, then obtain k and ε from the above velocity distributions, Eq. (9.60) and Eq. (9.61). Using these results, calculate G and F again, and then obtain 9.6 Comparison of Analyses Using the k-ε Model with Experiments 219 Fig. 9.10. Average velocity distribution of Couette flow [31] the pressure distribution again in the same way as in the beginning of the procedure. This calculation will be repeated until the calculated pressure distribution converges within a small error. As an example, consider a journal with a diameter of 150 mm rotating in a bear- ing with an inner diameter of 152 mm and a length of 150 mm over the speed range 100 – 5000 rpm. Let the coefficient of dynamic viscosity of the lubricating oil be 9.4 ×10 −6 m 2 /s(30 ◦ C). Comparisons of the theoretical and the experimental results, both under the above conditions unless otherwise stated, will be shown below. Lubricating oil is supplied at a rate of 12 l/min in the experiments. Figure 9.10 shows the theoretical and the experimental results of the average ve- locity distribution u for Couette flow, the experiment by Reichardt being used in this case [6]. It is seen in Fig. 9.10 that although the theoretical and the experimental re- sults are different for a Reynolds’ number of Re = 1200, they are in good agreement for Re = 2900 and Re = 34 000. This shows that the k-ε model is better suited to analysis in the turbulent region well above the laminar-to-turbulent transition region. Figure 9.11a,b shows the theoretical and experimental results for the pressure distribution in a finite width bearing. Figures 9.11a,ba,b are the nondimensional pres- sure distribution p in the circumferential direction at the center of bearing width for Re = 2000 and 8000, respectively, the parameter being the eccentricity ratios as shown. The theoretical and the experimental values are generally in good agreement. However, for Re = 8000 and an eccentricity ratio of 0.8, the experimental pressure [...]... a Full Circular Bearing and a Partial Arc Bearing in the Laminar and Turbulent Flow Regimes”, Journal of Lubrication Technology, Trans ASME, Series F, Vol 89, April 1967, pp 143 - 153 12 H.G Elrod, Jr and C.W Ng, “A Theory of Turbulent Fluid Films and its Application to Bearings”, Journal of Lubrication Technology, Trans ASME, Series F, Vol 89, No 3, 1967, pp 346 - 362 13 J.H Vohr, “An Experimental... Transactions, Vol 13, 1970, pp 262 - 268 17 H Aoki and M Harada, “Turbulent Lubrication Theory for Full Journal Bearings” (in Japanese), Journal of Japan Society of Lubrication Engineers, Vol 16, No 5, May 1971, pp 348 - 356 18 P Castle, F.R Mobbs and P.H Markho, “Visual Observations and Torque Measurements in the Taylor Vortex Regime Between Eccentric Rotating Cylinders”, Journal of Lubrication Technology,... The branching and merging may occur irregularly, resulting in somewhat disturbed vortices Based on the above observations, a local stability theory of the flow between two eccentric cylinders and a hydrodynamic lubrication theory taking the Taylor vortices into consideration were proposed [30] References 1 G.I Taylor, “Stability of a Viscous Liquid Contained Between Two Rotating Cylinders”, Phil Trans.,... Vol 36 (1956), pp S 26 - S 29 7 V.N Constantinescu, “On Turbulent Lubrication , Proceedings of the Institution of Mechanical Engineers, Vol 173, 1959, pp 881 - 900 8 R.C DiPrima, “A Note on the Stability of Flow in Loaded Journal Bearings”, ASLE Trans., Vol 6, 1963, pp 249 - 253 9 Chung-Wah Ng and C.H.T Pan, “A Linearized Turbulent Lubrication Theory”, Journal of Basic Engineering, Trans ASME, Series... experimental results of load capacity or Sommerfeld’s number are in good agreement even when the eccentricity ratio exceeds 0.95 This shows that the lubrication theory based on the k-ε model can be applied to bearings with very high eccentricity ratios Figure 9.13a,b shows the nondimensional pressure distributions p for the cases shown in Fig 9.12 The parameter for the curves is the nondimensional load... Study of Taylor Vortices and Turbulence in Flow Between Eccentric Rotating Cylinders”, Journal of Lubrication Technology, Trans ASME, Series F, Vol 90, No 1, 1968, pp 285 - 296 14 M Shirakura and H Ohashi, “Fluid Mechanics” (in Japanese), Corona Publishing Co., Ltd., Tokyo, 1969 15 C.M Taylor, “Turbulent Lubrication Theory Applied to Fluid Film Bearing Design”, Paper 6, Proceedings of the Institution... Nondimensional load capacity and Sommerfeld’s reciprocal versus eccentricity ratio [31] Fig 9.13a,b Nondimensional pressure distribution in a finite width bearing (2) for the two cases analyzed in Fig 9.12 [31] changes in Reynolds’ number do not very much affect the shape of the loci of the journal center 222 9 Turbulent Lubrication Fig 9.14 Locus of the journal center [31] 9.7 Reduction of Friction in a Turbulent... Two-Equation Model of Turbulence”, International Journal of Heat Mass Transfer, Vol 15, 1972, pp 301 - 313 22 W.P Jones and B.E Launder, “The Calculation of Low-Reynolds-Number Phenomena with a Two-Equation Model of Turbulence”, International Journal of Heat Mass Transfer, Vol 16, 1973, pp 1119 - 1130 23 H Aoki, T Kusaga, H Tada, H Sasaki, “A Study on Drag Reduction of Dilute Polymer Solutions in Annular... capacity P of Fig 9.12 The theoretical and experimental values for the pressure distribution are in good agreement even in the case of high eccentricity ratios over 0.9 This also shows that the turbulent lubrication theory based on the k-ε model can be used for bearings of high eccentricity ratio This is because the k-ε model is valid for large pressure gradients Figure 9.14 shows the theoretically obtained... Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers”, Proc of 1st International Congress on Rheology , North Holland Publishing Co., Amsterdam, Vol 2, 1949, pp 135 - 3 D.F Wilcock, “Turbulence in High-Speed Journal Bearings”, Trans ASME, Vol 72, August 1950, pp 825 - 834 References 227 4 J Laufer, “The Structure of Turbulence in Fully Developed Pipe Flow”, . ASLE Transactions, Vol 13, 1970, pp. 262 - 268. 17. H. Aoki and M. Harada, “Turbulent Lubrication Theory for Full Journal Bearings” (in Japanese), Journal of Japan Society of Lubrication Engineers,. ratio exceeds 0.95. This shows that the lubrication theory based on the k-ε model can be applied to bearings with very high eccentricity ratios. Figure 9.13a,b shows the nondimensional pressure. observations, a local stability theory of the flow between two eccen- tric cylinders and a hydrodynamic lubrication theory taking the Taylor vortices into consideration were proposed [30]. References 1.